Relativistic anisotropic hydrodynamics
In this paper we review recent progress in relativistic anisotropic hydrodynamics. We begin with a pedagogical introduction to the topic which takes into account the advances in our understanding of this topic since its inception. We consider both conformal and non-conformal systems and demonstrate how one can implement a realistic equation of state using a quasiparticle approach. We then consider the inclusion of non-spheroidal (non-ellipsoidal) corrections to leading-order anisotropic hydrodynamics and present the findings of the resulting second-order viscous anisotropic hydrodynamics framework. We compare the results obtained in both the conformal and non-conformal cases with exact solutions to the Boltzmann equation and demonstrate that, in all known cases, anisotropic hydrodynamics best reproduces the exact solutions. Based on this success, we then discuss the phenomenological application of anisotropic hydrodynamics. Along these lines, we review techniques which can be used to convert a momentum-space anisotropic fluid into hadronic degrees of freedom by generalizing the original idea of Cooper-Frye freeze-out to momentum-space anisotropic systems. And, finally, we present phenomenological results of 3+1d quasiparticle anisotropic hydrodynamic simulations and compare them to experimental data produced in 2.76 TeV Pb-Pb collisions at the LHC. Our results indicate that anisotropic hydrodynamics provides a promising framework for describing the dynamics of the momentum-space anisotropic QGP created in heavy-ion collisions.
keywords:quark-gluon plasma, hydrodynamics, anisotropic hydrodynamics, non-equilibrium dynamics
- 1 Introduction
2 Historical foundations and pedagogical introduction
- 2.1 Moments of the Boltzmann equation
- 2.2 Tensor basis
- 2.3 The conformal 0+1d aHydro distribution and bulk variables
- 2.4 The 0+1d equations of motion
- 2.5 Relation to second-order viscous hydrodynamics in the small anisotropy limit
- 2.6 Ideal and free streaming limits
- 2.7 aHydro numerical solution and comparison to vHydro
- 2.8 Summary
- 3 3+1d leading-order aHydro for non-conformal systems
- 4 3+1d quasiparticle anisotropic hydrodynamics
- 5 Second-order anisotropic hydrodynamics
- 6 The anisotropic matching principle
- 7 Testing against exact solutions to the Boltzmann equation
- 8 The anisotropic non-equilibrium attractor
- 9 Anisotropic Freeze-out
- 10 3+1d anisotropic hydrodynamics phenomenology
- 11 Conclusions and outlook
The ongoing ultrarelativistic heavy-ion collision (URHIC) experiments at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) and the Large Hadron Collider (LHC) at the European Organization for Nuclear Research (CERN) aim to produce and study the properties of the quark-gluon plasma (QGP) using a variety of collision systems, e.g. AA, pA, dA, and pp over a wide range of center-of-mass energies. One of the surprising findings of the experiments at RHIC was that the collective behavior of the soft hadrons GeV was qualitatively well-described by ideal relativistic hydrodynamics Huovinen:2001cy (); Hirano:2002ds (); Kolb:2003dz (). In these early days, due to the fact that ideal hydrodynamics implicitly relies on an assumption that the system is in perfect isotropic thermal equilibrium, this led to the widespread supposition that the QGP created in URHICs underwent fast isotropization/thermalization on a timescale fm/c (see e.g. Kolb:1999it (); Heinz:2001ax (); Heinz:2002un (); Jacobs:2004qv (); Muller:2005en (); Strickland:2007fm ()).
Despite subsequent relativistic hydrodynamics research Muronga:2001zk (); Muronga:2003ta (); Muronga:2004sf (); Heinz:2005bw (); Baier:2006um (); Romatschke:2007mq (); Baier:2007ix (); Dusling:2007gi (); Luzum:2008cw (); Song:2008hj (); Heinz:2009xj (); El:2009vj (); PeraltaRamos:2009kg (); PeraltaRamos:2010je (); Denicol:2010tr (); Denicol:2010xn (); Schenke:2010rr (); Schenke:2011tv (); Bozek:2011wa (); Niemi:2011ix (); Niemi:2012ry (); Bozek:2012qs (); Denicol:2012cn (); Denicol:2012es (); PeraltaRamos:2012xk (); Jaiswal:2013npa (); Jaiswal:2013vta (); Calzetta:2014hra (); Denicol:2014vaa (); Denicol:2014mca (); Jaiswal:2014isa (), which included the effect of dissipative corrections in the form of second-order viscous hydrodynamics (vHydro) and explicitly included pressure anisotropies in the form of the viscous stress tensor, it was hard to dislodge the idea that the QGP generated in URHICs was in perfect isotropic equilibrium. Simultaneously to these phenomenological developments, fundamental studies of the thermalization and isotropization of the QGP were undertaken in the context of quantum field theories in both the weak Heinz:1985vf (); Mrowczynski:1988dz (); Pokrovsky:1988bm (); Mrowczynski:1993qm (); Blaizot:2001nr (); Romatschke:2003ms (); Arnold:2003rq (); Arnold:2004ih (); Romatschke:2004jh (); Arnold:2004ti (); Mrowczynski:2004kv (); Rebhan:2004ur (); Rebhan:2005re (); Romatschke:2005pm (); Romatschke:2006nk (); Romatschke:2006wg (); Rebhan:2008uj (); Fukushima:2011nq (); Kurkela:2011ti (); Kurkela:2011ub (); Blaizot:2011xf (); Attems:2012js (); Kurkela:2012tq (); Berges:2012iw (); Blaizot:2013lga (); Mrowczynski:2016etf () and strong coupling Chesler:2008hg (); Grumiller:2008va (); Chesler:2009cy (); Albacete:2009ji (); Wu:2011yd (); Heller:2011ju (); Chesler:2011ds (); Heller:2012je (); vanderSchee:2012qj (); Romatschke:2013re (); Casalderrey-Solana:2013aba (); vanderSchee:2013pia () limits. The conclusion of these studies was that, due to the extreme conditions in which the QGP is created, it is not possible to achieve isotropization on a sub-fm/c timescale. This motivated the investigation of the impact of momentum-space anisotropies on QGP dynamics and signatures and, after many years, there is now a consensus in the theoretical community that the QGP possesses a high degree of momentum-space anisotropy at early times and near the dilute edges of the system Ryblewski:2013jsa (); Strickland:2014pga (). In practice, one finds that, in these spacetime regions, the transverse pressure in the local rest frame greatly exceeds the longitudinal pressure and that, in the center of the fireball, it takes many fm/c for the system to become even approximately isotropic. Faced with this, researchers began looking for ways to formulate hydrodynamics in a momentum-space anisotropic QGP.
A significant breakthrough occurred in this direction with two papers, one from Florkowski and Ryblewski Florkowski:2010cf () and the other from Martinez and Strickland Martinez:2010sc () in 2010. In both papers, the authors considered a boost-invariant and transversally homogeneous system but in the first paper the authors postulated an equation governing entropy production in an anisotropic system, whereas in the second one the authors took moments of the Boltzmann equation in relaxation-time approximation. These papers demonstrated that it was possible to formulate relativistic hydrodynamics using an intrinsically anisotropic background. In addition, the Martinez and Strickland paper demonstrated that (a) the resulting dynamical equations could reproduce both the ideal hydrodynamics limit () and the free-streaming limit () and (b) that, in the case of weak momentum space isotropy (), the formalism reproduced standard viscous hydrodynamics. This framework has become known, generally, as anisotropic hydrodynamics (aHydro). The original ideas presented in Refs. Florkowski:2010cf (); Martinez:2010sc () have since been generalized and extended to fully 3+1d systems with broken conformal symmetry. These generalized frameworks have been applied successfully to heavy-ion collision phenomenology (see e.g. Ryblewski:2010ch (); Florkowski:2011jg (); Martinez:2012tu (); Ryblewski:2012rr (); Bazow:2013ifa (); Tinti:2013vba (); Nopoush:2014pfa (); Tinti:2015xwa (); Bazow:2015cha (); Strickland:2015utc (); Alqahtani:2015qja (); Molnar:2016vvu (); Molnar:2016gwq (); Alqahtani:2016rth (); Bluhm:2015raa (); Bluhm:2015bzi (); Alqahtani:2017jwl (); Alqahtani:2017tnq ()).
In this review, rather than directly presenting the modern version of aHydro, we will attempt to make historical review, highlighting the important advances in stages in order to make the material more accessible to readers who are unfamiliar with the literature. We begin with a pedagogical introduction to the topic which takes into account the advances in our understanding of this topic since its inception. We consider both conformal and non-conformal systems and demonstrate how one can implement a realistic equation of state using a quasiparticle approach. We then consider the inclusion of non-spheroidal (non-ellipsoidal) corrections to leading-order anisotropic hydrodynamics and present the findings of the resulting second-order viscous anisotropic hydrodynamics framework. We compare the results obtained in both the conformal and non-conformal cases with exact solutions to the Boltzmann equation and demonstrate that, in all known cases, anisotropic hydrodynamics best reproduces the exact solutions. Based on this success, we then discuss the phenomenological application of aHydro. Along these lines, we review techniques which can be used to convert a momentum-space anisotropic fluid into hadronic degrees of freedom by generalizing the original idea of Cooper-Frye freeze-out to momentum-space anisotropic systems. And, finally, we present phenomenological results of 3+1d quasiparticle anisotropic hydrodynamic simulations and compare them to experimental data produced in 2.76 TeV Pb-Pb collisions at the LHC. Our results indicate that aHydro provides a promising framework for describing the dynamics of the momentum-space anisotropic QGP created in heavy-ion collisions.
Conventions and notation
Unless otherwise indicated, the Minkowski metric tensor is taken to be “mostly minus”, i.e. . The transverse projection operator is used to project four-vectors and/or tensors into the space orthogonal to . Parentheses and square brackets on indices denote symmetrization and anti-symmetrization, respectively, i.e. and . Angle brackets on indices indicate projection with a four-index transverse projector, , where projects out the traceless and -transverse components of a rank-two tensor.
2 Historical foundations and pedagogical introduction
In this section, we review the method presented originally in Ref. Martinez:2010sc () in a concise and updated manner. At the end, we will highlight the important findings. We will attempt to present the material in a pedagogical manner so as to provide the reader a firm basis to build upon. As in the original paper, we will make several simplifying assumptions: (1) that the system is invariant under longitudinal boosts, (2) that the system is transversally homogenous, and (3) that the system is conformal (massless particles). Hence, we consider here, a conformal 0+1d system. In the course of this review, we will relax all of these assumptions.
2.1 Moments of the Boltzmann equation
The starting point for the derivation presented in Ref. Martinez:2010sc () was the relativistic Boltzmann equation
where and are the particle four-momentum and -position, respectively, is the one-particle distribution function, and is the collisional kernel which includes both elastic and inelastic scatterings to all orders. For this discussion, we will assume that the system is boost-invariant and transversally homogenous (0+1d) and that there is no chemical potential.
To proceed, one can take moments of Eq. (1) using the integral operator
is the Lorentz-invariant momentum integration measure with being the number of degrees of freedom (degeneracy). Acting with on Eq. (1), one obtains the “zeroth-moment” of the Boltzmann equation
Using the fact that the particle current is defined as
and defining a general moment of the collisional kernel via
we can write the zeroth moment compactly as
Repeating this exercise using , one obtains the first-moment of the Boltzmann equation
is the energy-momentum tensor. For any valid microscopic model, the energy and momentum conserving delta function inherent in the collisional kernel ensures that . If, instead, one works with an effective collisional kernel such as the relaxation-time approximation (RTA) model, the requirement that becomes a constraint which must be enforced on any parameters appearing in the model’s kernel. We will return to this point later. Hence one finds, in general, that the first moment of the Boltzmann equation results in the simple statement of energy and momentum conservation
In Ref. Martinez:2010sc () the zeroth and first moments were used to obtain the necessary equations of motion, however, in order to bring the material up to date, we would like to also compare a different prescription in which one uses a linear combination of equations obtained from the second moment of the Boltzmann equation Tinti:2013vba (). For this purpose, we note that, for a general moment, one obtains
2.2 Tensor basis
At this point in the discussion it is helpful to write down the most general forms for the particle current , energy-momentum tensor , and third-rank tensor . For this purpose, we need only consider the symmetries of the system. To begin, we introduce four 4-vectors which span spacetime in the local rest frame (LRF) Florkowski:2011jg (); Martinez:2012tu ():
These four-vectors are orthonormal in all frames. The vector is associated with the four-velocity of the LRF and is conventionally called . One can also identify , , and as indicated above.111In the lab frame, the three space-like vectors can be written entirely in terms of . This is because can be obtained by a sequence of Lorentz transformations/rotations applied to the LRF expressions specified above Florkowski:2011jg (); Martinez:2012tu ().
The metric tensor can be expressed in terms of these four-vectors as
where the sum extends over . In addition, the standard transverse projection operator, which is orthogonal to , can be expressed in terms of the basis (12)
from which one finds . The space-like components of the tensor basis are eigenvectors of this operator, i.e. .
Using these basis vectors, we can expand any tensor. For example, the number current, which is a rank-1 tensor, can be written in general as
Since the 0+1d distribution function is reflection symmetric in momentum-space around the , , and directions in the LRF, the space-like coefficients , leaving in this case
Likewise, since the the energy-momentum tensor is symmetric , one has
where the coefficients , etc. are scalar fields. Using the symmetries associated with the 0+1d case considered in this section, one can simplify this to Martinez:2012tu ()
where , , and are the energy density, transverse pressure, and longitudinal pressure in the LRF. Above, and correspond to the directions parallel and transverse to the anisotropy direction , respectively.
Finally, we can repeat this exercise for the rank-three tensor
where we have already made use of the 0+1d symmetries. All other possible combinations vanish by symmetry. and can be found by appropriate projections, for example, can be obtained by taking of as follows
with . We note that, for the 0+1d case, the corresponding basis vectors in the lab frame can be obtained using a longitudinal boost with the boost velocity equal to the spatial rapidity since the system is boost-invariant. This gives
We will make use of these forms in the forthcoming discussion.
2.3 The conformal 0+1d aHydro distribution and bulk variables
To proceed one must specify a form for the distribution function which appears in the various integrals above. In anisotropic hydrodynamics one allows the one-particle distribution function in the LRF to be intrinsically momentum-space anisotropic. For this introductory presentation, we are considering a 0+1d system, in which case it suffices to introduce a single momentum-space anisotropy parameter , as follows,
where indicates the Romatschke-Strickland form Romatschke:2003ms (); Romatschke:2004jh (), is an equilibrium distribution function, is the local anisotropy parameter and is the local scale parameter which reduces to the temperature in the isotropic limit, . Note that all momenta appearing above are specified in the LRF of the system. We will formulate this in an explicitly Lorentz-invariant manner in a forthcoming section, but for now we will continue with this form.
Using this form, it is possible to evaluate the necessary moments of the distribution function analytically. For example, in the conformal case considered in this section, the number density in the LRF can be factorized into a function that depends solely on and another function which depends solely on the scale
where, in order to evaluate the integral, we have made a change of variables to , then relabeled , and recognized that the remaining integral is nothing by the isotropic number density evaluated at the momentum scale . Similarly, the components of the energy-momentum tensor can be evaluated analytically, e.g. the LRF energy density can be factorized
Performing the angular integration and repeating this exercise for the transverse and longitudinal pressures given by and , respectively, one obtains
which satisfy . This follows from the fact that the conformal energy-momentum tensor is traceless, .
Turning, finally to the rank–three tensor, using the 0+1d aHydro distribution function, one finds
which satisfy .
2.4 The 0+1d equations of motion
where is the co-moving derivative and is the expansion scalar. Using the 0+1d basis vector (21) and transforming to Milne coordinates using and , one obtains
In order to reach the final form we will need to specify the collisional kernel. Following Ref. Martinez:2010sc () we will assume that the collisional kernel is given by the RTA form
where is the relaxation time, which must be inversely proportional to the local (effective) temperature T in the conformal case and is the shear viscosity to entropy density ratio.
As mentioned previously, in order to conserve energy and momentum it is necessary that the first moment of the collisional kernel vanish, i.e. . This is trivially satisfied for in RTA due to the symmetries of and and for it results in the so-called Landau matching condition
Using this and Eq. (26), in the 0+1d conformal case, one obtains
The temperature determined in this manner will be called the effective temperature. In the end it is a stand-in for the local energy density which is well-defined both in and out of equilibrium.
Evaluating using Eq. (23), one obtains
Performing a similar manipulation on the first moment equation, starting from the simplified form of energy conservation in 0+1d, i.e.
Note that the three equations related to momentum conservation are, once again, automatically satisfied if the distribution function is reflection symmetric in momentum space.
Finally, turning to the second moment, the projection of Eq. (11) with gives
and the and projections both give
Combining the projection minus one-third of the sum of the , , and projections gives
Summarizing, from the zeroth, first, and second moments of the Boltzmann equation we obtain three equations of motion given by Eqs. (35), (37), and (40). Of course, we can continue in this manner ad-infinitum to the third moment, etc., however, in practice we only need two equations of motion to evolve and . Since energy-momentum conservation is sacrosanct, it must be included in the set. In addition, since higher moments are sensitive to high-momentum behavior of the distribution and we are looking for equations that describe the long wavelength dynamics, we are guided naturally to consider the lowest possible momentum-moments. Therefore, based on the equations presented thus far, that leaves two possibilities: (a) zeroth+first and (b) first+second. In order to decide which option to use in practice, ones needs to consider the near equilibrium (small anisotropy) limit. One can show that the option (a) does not reproduce the correct near-equilibrium equations if one uses ; however, as we will demonstrate in the next subsection, option (b) automatically reproduces the correct near-equilibrium limit Tinti:2013vba ().222In Ref. Martinez:2010sc () the authors used option (a), but fixed it by hand by adjusting the relaxation time by a factor two in order to match Israel-Stewart theory in the near-equilibrium limit.
2.5 Relation to second-order viscous hydrodynamics in the small anisotropy limit
In order to make the connection to standard second-order vHydro, one can rewrite Eq. (40) in terms of the single shear stress tensor component necessary for a conformal 0+1d system, i.e. and . To start, we note that the energy conservation equation (36) can be expressed in terms of as
To relate and one can use , to obtain
In the left panel of Fig. 1 we plot as a function of determined via Eq. (42) and, in the right panel, we plot as a function of determined via numerical inversion of Eq. (42). Importantly, one observes that in aHydro is bounded, . This is related to the positivity of the longitudinal and transverse pressures which naturally emerges in this framework. Furthermore, one can easily show that is related to the shear inverse Reynolds number via Denicol:2012cn ()
from which one can see that the shear inverse Reynolds number measures the relative magnitude of the shear viscous correction to the energy-momentum tensor compared to the isotropic pressure. We also note that, as a consequence of Eq. (43), a series in can be loosely understood as a series in .
and the understanding that with being the inverse function of (shown in the right panel of Fig. 1). Written in this form, we can see explicitly that the aHydro second-moment equation sums an infinite number of terms in the expansion in the inverse Reynolds number (43). Next, we will expand this equation in powers of the inverse Reynolds number through second order and compare it to standard vHydro.
In order to complete the connection to standard vHydro, one can expand Eq. (46) in around . For this purpose we need the expansions of the various functions that are necessary. At second-order in , one finds
Inverting the relationship between and to second-order in gives
which results in
Using this expansion in Eq. (46) and keeping terms through linear order in gives
where, on the left hand side, we have used the fact that one can eliminate the energy density by expressing it in terms of the transport coefficients
and relabeled in order to cast the equations in “standard” second order hydrodynamics form. Equation (51) agrees with previously obtained RTA second-order vHydro results Denicol:2010xn (); Denicol:2012cn (); Denicol:2014loa (); Jaiswal:2013vta (); Jaiswal:2013npa (), demonstrating that, in the limit of small momentum-space anistropy, aHydro automatically reproduces the correct second-order vHydro equations if one uses the first and second moments of the Boltzmann equation to generate the dynamical equations.
2.6 Ideal and free streaming limits
In the previous subsection we proved that, using the first and second moments of the Boltzmann equation, aHydro reduces to vHydro (vHydro) in the small anisotropy limit. In RTA, the small anisotropy limit is appropriate when the relaxation time of the system is very small, which occurs in the limit that (or ). From the previous subsection it is easy to see that when and hence , the aHydro equations reproduce the equation of motion of ideal hydrodynamics Martinez:2010sc (). Importantly, however, the equations also contain the free streaming (FS) limit. To see this, we consider the opposite limit, namely the limit , which corresponds to .
Taking the limit in Eq. (40) one obtains
which has a solution of the form
which tells us that in the FS limit . These solutions for and correspond precisely with the analytic result for the case of 0+1d free streaming Baym:1984np (); Mauricio:2007vz (); Martinez:2009mf (); Martinez:2009ry (). The fact that aHydro can reproduce the ideal limit, the free streaming limit, and second-order vHydro in the limit of small makes it a unique approach to dissipative dynamics. Typically, one must rely on arguments based on near-equilibrium limits to obtain fluid dynamical equations, however, aHydro shows that it may also be possible to describe certain classes of far-from-equilibrium dynamics using an optimized fluid-dynamical approach.
2.7 aHydro numerical solution and comparison to vHydro
Next, we turn to the numerical solution of the 0+1d conformal aHydro and vHydro equations. In Fig. 2 we plot the effective temperature and pressure anisotropy () as a function of proper time (). For these figures we used an initial condition of MeV, , fm/c. The solid lines were generated using Eqs. (37) and (40) with and the effective temperature given by Eq. (33). The dashed lines were generated using Eqs. (35) and (37) with using the original prescription in Ref. Martinez:2010sc (). As can be seen from the left panel the temperature evolution predicted by both aHydro schemes is quite similar, however, the pressure anisotropy predicted by each scheme is quantitatively different, particularly at large values of . That being said, comparing the qualitative aspects we find similar predictions from both schemes. Since the scheme which uses 1st and 2nd moments automatically reproduces the near-equilibrium limit it is natural to use this scheme if one uses the method of moments of the Boltzmann equation.
In Fig. 3, we compare the 0+1d pressure anisotropy as a function of proper time predicted by the aHydro 1st and 2nd moment scheme and Denicol-Niemi-Molnar-Rischke (DNMR) vHydro Denicol:2012cn () for MeV (left) and MeV (right) with 0.25 fm/c. For DNMR vHydro we numerically solved Eqs. (41) and (51). DNMR vHydro is a complete second-order treatment which is based on kinetic theory. In both panels of Fig. 3 we took the system to be initially isotropic in momentum space, corresponding to and for aHydro and vHydro, respectively. As one can see from this figure, aHydro and vHydro agree qualitatively concerning the magnitude of the pressure anisotropy. Comparison of the left and right panels demonstrates that lower initial temperatures result in larger deviations from isotropy. This is to be expected since the relaxation time scales inversely with the local effective temperature. Finally, we note that there are sizable quantitative differences between aHydro and vHydro evolutions. Importantly, we note that in both panels, the vHydro pressure ratio is observed to go negative for large values of . This is indicative of a complete breakdown of vHydro which is not surprising when is large since traditional vHydro are based on a linearization around isotropic equilibrium and the corrections are proportional to this ratio. We note, however, that at even lower temperatures one sees this breakdown for even small values of . Finally, we note that apart from this apparent violation of positively of the longitudinal pressure, it’s hard to say whether aHydro or vHydro provide a more quantitatively reliable description of the system’s evolution. In a forthcoming section, we will compare the various schemes developed with exact solutions of the Boltzmann equation which are available in some simple situations. From these comparisons we will learn that the aHydro framework provides the most quantitatively reliable method.
In this section, we presented the basic ingredients of the aHydro formalism in the case of a conformal system which is transversally homogeneous and boost invariant (0+1d). We demonstrated that aHydro reproduces the ideal hydrodynamics and free streaming limits. In addition, we demonstrated that it reproduces the correct equations of second-order vHydro in the limit of small anisotropy. This sets the stage for (a) extending the formalism to 3+1d by relaxing all of the symmetries assumed in the prior subsection and (b) applying the formalism to non-conformal (massive) gases. In the next section, we will do this at leading order in the aHydro expansion.
3 3+1d leading-order aHydro for non-conformal systems
In this section we will present 3+1d aHydro for a non-conformal QGP at leading-order in the aHydro expansion. By “leading-order” we simply mean that we take into account possible momentum-space anisotropies by allowing for a generalized ellipsoidal form for the one-particle distribution function and ignore any possible deviations from the assumed form. In aHydro, one assumes that the full one-particle distribution function is given by a leading-order term of generalized Romatschke-Strickland form Romatschke:2003ms (); Romatschke:2004jh () plus a correction term which accounts for deviations from the generalized ellipsoidal form
where is an energy scale which becomes the temperature in the isotropic equilibrium limit and is the chemical potential.333We have called the scale here to emphasize that it not necessarily equal to the momentum scale associated with the canonical Romatschke-Strickland form used in the previous section. The explicit relation between the two scales can be found in Sec. IIC of Ref. Nopoush:2014pfa (). The anisotropy tensor has the form where is a symmetric traceless tensor obeying and , is the bulk degree of freedom, and is the transverse projector defined in the conventions and notation block in the beigging of the review Martinez:2012tu (); Nopoush:2014pfa (). Using the ellipsoidal form (56) and the tracelessness of , we are left with six independent parameters out of the seven original parameters , , and . Combining these six with the three independent parameters which describe and the one for the momentum scale , we arrive at ten degrees of freedom, which suffice to describe the dynamics of the ten independent components of the energy-momentum tensor. In thermal equilibrium, the distribution function can be identified as Fermi-Dirac, Bose-Einstein, or Maxwellian distribution. Unless otherwise indicated, we will take the Boltzmann form. Finally, in this section, following the leading-order aHydro model, we ignore the non-ellipsoidal deviations accounted for by . We will return to this issue in Sec. 5 where we will discuss second-order aHydro (dubbed vaHydro in the literature) in which one uses orthonormal polynomial expansions of similar to standard vHydro to compute these corrections systematically.
In what follows in this section, we will consider a 3+1d system consisting of particles with a temperature independent mass following Ref. Nopoush:2014pfa (). The introduction of the mass scale will allow us to study bulk viscous correction in the context of aHydro. For simplicity, we will assume vanishing chemical potential herein.
3.1 Basis vectors
As mentioned previously, the lab frame basis vectors for a general 3+1d system can be obtained by a set of Lorentz transformations applied to the LRF basis vectors Martinez:2012tu (); Ryblewski:2010ch (). The set of Lorentz transformations correspond to a longitudinal boost by along the beam line, a rotation by around the beam line, and a transverse boost , which together yield
where . Note that these basis vectors reduce to the 0+1d form specified in Eq. (21) under the assumption of transverse homogeneity ( followed by ) and boost invariance ().
3.2 Diagonal ellipsoidal form
In order to simplify the formalism, in this section we will make an additional assumption, namely that the anisotropy tensor is diagonal, i.e. . This assumption is exact for central collisions. For non-central collisions, results from standard vHydro simulations suggest that the off-diagonal components of the anisotropy are small Song:2009gc (), which suggests that one can ignore the off-diagonal anisotropies to first approximation. In forthcoming sections (Secs. 6 and 5, respectively), we will discuss how to relax this assumption in the context of leading- and second-order aHydro.
Due to the tracelessness of the tensor, one has . Expanding the argument of the square root appearing in Eq. (56) in the LRF gives
where and we have introduced some more convenient anisotropy parameters
This set of three anisotropy parameters replace the two independent components of and the degree of freedom . Using Eq. (59) and , one has
3.3 Dynamical equations
Assuming a diagonal anisotropy tensor, the energy-momentum tensor can be expressed as
The associated energy density and pressures can be expressed as
with and the -functions are
where , , , , , and . The and functions appearing above are
More details concerning the -functions and their efficient evaluation can be found Refs. Nopoush:2014pfa (); Alqahtani:2015qja (); Alqahtani:2016rth (); Alqahtani:2017tnq (). Note that, in the isotropic limit, , and assuming Boltzmann statistics, one has and
3.3.1 First moment
The first moment of Boltzmann equation results in four equations